Change of basis, equation of a line

A line has the equation in the standard basis of R2. The question is what is the equation in the basis given by the matrix .

I'm used to doing this to vectors, so what I tried is to solve the equation

The basis matrix is invertible, so

But that is all wrong. Apparently what I want to do is multiply with the transpose of the vector. Multiplying the vector to the transposed matrix also works. (So I guess my method would produce the right answer for an orthogonal matrix.)

But why? Why can't I take a vector to represent the normal of the line, change its basis and get the normal of the line represented in the new basis?

Thanks in advance for any help, and apologies if this is a too easy question that should go in some other section.

Have you learned about coordinate matrices and the matrix of a linear map? If so, there is a simple, but deep, formula you should have learned:

where are the standard basis and the other basis, respectfully, and is the matrix associated with the linear map (and depending on book and instructor, the might be switched on the super-/subscript of ... this is how I learned to write it out).

Unfortunately no. Change of basis comes very early in the chapter about linear transformations/maps. I've scoured the textbook but I can't find anything resembling that fomula. Where does it come from? Does it require knowledge of eigenvalues? (which I understand can come before transformations but is actually the chapter after in our book.)

To clarify, I can solve the problem. What I lack is some sort of intuition of why I can't treat the equation as the vector and manipulate it to get a correct answer. It feels like it is something very basic I've missed.